Matrix algebra from one of the students Fall '00

This is my way of simplifying thinking about matrices:

First of all, think of a very simple example that does not need a matrix

to solve it:

You want to see if the following sequence: ATTG exactly matches a second

sequence: ATTG, and to do that give a score of 1 for a match and 0 for

mismatch.

For each sequence, we will compare the first

bases to each other, and the second bases to each other, and so on. This

is called string comparison, so we get:

ATTG

ATTG

1 1 1 1

Now consider that we want to find an exact match for ATTG in the following

sequence:

CCATTGCC

In this case, we will have to compare the first base in the first sequence

with each base in the second sequence, then compare the second base in the

first sequence to each base in the second sequence, and so on. The best

way to present this is in the form of a table:, where we can put one

sequence in a raw, and the second in a column:

A T T G

C 0 0 0 0

C 0 0 0 0

A 1 0 0 0

T 0 1 0 0

T 0 0 1 0

G 0 0 0 1

C 0 0 0 0

C 0 0 0 0

But look, it is a matrix comparison. As mentioned in the class, do not

think of algebra for this matrix. Matrix here is used as a form of

comparison.

The reason for using the terms i, and j is that, considering the above

example, we have 2 sequences: N and M. the length of sequence N is 4 (n =

4), and the length of sequence M is 8 (m = 8). To refer to the first base

in the sequence M use M1. To refer to the fourth base in sequence N, use

N4. To refer to the score of comparing M1 and N4, that would be the

element S1,4 in the matrix, i.e. each element in the matrix is represented

by Si,j, where i = 1 to n (which is 4), and j = 1 to m (which is 8).

So looking at the matrix:

Sequence M is: M1 M2 M3 M4

Sequence N is:

N1 S1,1 S2,1 S3,1 S4,1

N2 S1,2 S2,2 S3,2 S4,2

N3 S1,3 S1,3 S3,3 S4,3

.

.

.

If S3,2 is Si,j, then Si-1,j is S2,2, and Si-1,j-1 is S2,1 and so on.

So we use i and j to help us and the computer to refer to specific

elements in the matrix.

So that is the way you think in terms of matrices.

Reema